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Maths

Probability - Higher

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# AND/OR rule

When two events are independent, the outcome of one has no effect on the outcome of the other.

When two events are said to be mutually exclusive they cannot happen at the same time. For example, if a die is thrown, the events 'obtaining a 6' and 'obtaining a 1' are mutually exclusive, as they cannot happen at the same time.

The events 'obtaining a six' and 'obtaining an even number' are not mutually exclusive, because throwing a six fits into both categories.

When two events, A and B, are independent:

P(A and B) = P(A) x P(B)

When two events, A and B, are mutually exclusive:

P(A or B) = P(A) + P(B)

Notice how the word and has been replaced by a multiplication sign for independent events, and the word or has been replaced by an addition sign for mutually exclusive events. This forms the basis of the AND/OR rule.

## Example

A bag contains 5 green beads and 4 red beads. A bead is taken from the bag, its colour noted, and it is then replaced. A second bead is then taken from the bag. What is the probability that the two beads are different colours?

We are being asked to find P (1st is green and 2nd is red or 1st is red and 2nd is green).

The 1st bead is replaced before the 2nd bead is taken out, so the first and second beads are independent. Therefore, the word and can be replaced by a multiplication sign.

The events '1st is green and 2nd is red', or '1st is red and 2nd is green', are mutually exclusive. Therefore, the word or can be replaced by an addition sign.

The answer to the question is

• 5/9 × 4/9 + 4/9 × 5/9
• = 20/81 + 20/81
• = 40/81

There are many pitfalls to using the AND/OR rule in this way. Only do so if you are very confident about the method and the question does not require the use of a tree diagram.

However, remembering the AND/OR rule will make sure you know when to add and when to multiply when calculating probabilities from tree diagrams.

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